Gell-Mann matrices
The Gell-Mann matrices, developed by , are a set of eight 3×3 used in the study of the in . They span the of the group in the defining representation. Matrices : and g_i = \lambda_i/2 . Properties These matrices are , Hermitian (so they can generate group elements through exponentiation), and obey the extra trace orthonormality relation. These properties were chosen by Gell-Mann because they then naturally generalize the for to , which formed the basis for Gell-Mann's . Gell-Mann's generalization further . For their connection to the of Lie algebras, see the . Trace orthonormality In mathematics, orthonormality typically implies a norm which has a value of unity (1). Gell-Mann matrices, however, are normalized to a value of 2. Thus, the of the pairwise product results in the ortho-normalization condition : \operatorname{tr}(\lambda_i \lambda_j) = 2\delta_{ij}, where \delta_{ij} is the . This is so the embedded Pauli matrices corresponding to the three embedded subalgebras of ''SU(2) are conventionally normalized. In this three-dimensional matrix representation, the is the set of linear combinations (with real coefficients) of the two matrices \lambda_3 and \lambda_8 , which commute with each other. There are three independent subalgebras: * \{\lambda_1, \lambda_2, \lambda_3\} * \{\lambda_4, \lambda_5, x\}, and * \{\lambda_6, \lambda_7, y\}, where the and are linear combinations of \lambda_3 and \lambda_8 . The SU(2) Casimirs of these subalgebras mutually commute. However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations. Commutation relations The 8 generators of SU(3) satisfy the : \begin{align} \left[ \lambda_a, \lambda_b \right] &= 2 i \sum_c f^{abc} \lambda_c, \\ \{ \lambda_a, \lambda_b \} &= \frac{4}{3} \delta_{ab} I + 2 \sum_c d^{abc} \lambda_c, \end{align} with the s : \begin{align} f^{abc} &= -\frac{1}{4} i \operatorname{tr}(\lambda_a [ \lambda_b, \lambda_c ]), \\ d^{abc} &= \frac{1}{4} \operatorname{tr}(\lambda_a \{ \lambda_b, \lambda_c \}). \end{align} The s f^{abc} are completely antisymmetric in the three indices, generalizing the antisymmetry of the \epsilon_{jkl} of . For the present order of Gell-Mann matrices they take the values : f^{123} = 1 \ , \quad f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = \frac{1}{2} \ , \quad f^{458} = f^{678} = \frac{\sqrt{3}}{2} \ . In general, they evaluate to zero, unless they contain an odd count of indices from the set {2,5,7}, corresponding to the antisymmetric (imaginary) s. Using these commutation relations, the product of Gell-Mann matrices can be written as : \lambda_a \lambda_b = \frac{1}{2} (\lambda_a,\lambda_b + \{\lambda_a,\lambda_b\}) = \frac{2}{3} \delta_{ab} I + \sum_c \left(d^{abc} + i f^{abc}\right) \lambda_c , where I'' is the identity matrix. Fierz completeness relations Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierz 'completeness relations', (Li & Cheng, 4.134), analogous to that . Namely, using the dot to sum over the eight matrices and using Greek indices for the their row/column indices, the following identities hold, : \delta^\alpha _\beta \delta^\gamma _\delta = \frac{1}{3} \delta^\alpha_\delta \delta^\gamma _\beta +\frac{1}{2} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta and : \lambda^\alpha _\beta \cdot \lambda^\gamma _\delta = \frac{16}{9} \delta^\alpha_\delta \delta^\gamma _\beta -\frac{1}{3} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta ~. One may prefer the recast version, resulting from a linear combination of the above, : \lambda^\alpha _\beta \cdot \lambda^\gamma _\delta = 2 \delta^\alpha_\delta \delta^\gamma _\beta -\frac{2}{3} \delta^\alpha_\beta \delta^\gamma _\delta ~. Representation theory A particular choice of matrices is called a , because any element of SU(3) can be written in the form \mathrm{exp}(i \theta^j g_j) , where the eight \theta^j are real numbers and a sum over the index is implied. Given one representation, an equivalent one may be obtained by an arbitrary unitary similarity transformation, since that leaves the commutator unchanged. The matrices can be realized as a representation of the s of the called . The of this group (a real Lie algebra in fact) has dimension eight and therefore it has some set with eight generators, which can be written as g_i , with ''i taking values from 1 to 8. Casimir operators and invariants The squared sum of the Gell-Mann matrices gives the quadratic , a group invariant, : C = \sum_{i=1}^8 \lambda_i \lambda_i = \frac{16} 3 I where I\, is 3×3 identity matrix. There is another, independent, , as well. Application to These matrices serve to study the internal (color) rotations of the s associated with the coloured quarks of (cf. ). A gauge colour rotation is a spacetime-dependent SU(3) group element U=\exp (i \theta^k ({\mathbf r},t) \lambda_k/2) , where summation over the eight indices is implied. References Category:Advanced mathematics